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<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="o_ring_1.html#NR2" target="_self">is_a_Sum_of_squares</a> 
 <b>and </b><br/><a NAME="E2:65"/><i><font color="Green">E30</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a>  );<br/>
<a NAME="E4:65"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E5:65"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E34</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> implies <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a>  )
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:65_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:65_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E4:65_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="o_ring_1.html#D9" target="_self">Def9</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:65_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR1" target="_self">is_a_square</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:65_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:65_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:65_1_2"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="vectsp_1.html#NM3">Scalar</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">y</font> <a href="o_ring_1.html#NR1" target="_self">is_a_square</a>  &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="o_ring_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="txt" href="o_ring_1.html#E20"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:65_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 <b>by </b><i><a class="ref" href="o_ring_1.html#D8" target="_self">Def8</a>, <a class="ref" href="o_ring_1.html#D9" target="_self">Def9</a>, <a class="txt" href="o_ring_1.html#E22"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="o_ring_1.html#D5" target="_self">Def5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:65_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 <b>by </b><i><a class="txt" href="o_ring_1.html#E21"><i><font color="Green">Lemma59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:65"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="o_ring_1.html#E16"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="o_ring_1.html#E17"><i><font color="Green">Lemma55</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:65"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="o_ring_1.html#NR13" target="_self">is_generated_from_squares</a> 
 <b>by </b><i><a class="ref" href="o_ring_1.html#D6" target="_self">Def6</a>, <a class="ref" href="o_ring_1.html#D7" target="_self">Def7</a></i>;<br/>


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